Properties

Label 4016.2.a.m.1.16
Level 4016
Weight 2
Character 4016.1
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 23
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(23\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) = 4016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.19399 q^{3} -1.76720 q^{5} +4.20476 q^{7} -1.57440 q^{9} +O(q^{10})\) \(q+1.19399 q^{3} -1.76720 q^{5} +4.20476 q^{7} -1.57440 q^{9} +6.20169 q^{11} +6.78297 q^{13} -2.11002 q^{15} -4.29929 q^{17} +2.98396 q^{19} +5.02042 q^{21} +4.21325 q^{23} -1.87699 q^{25} -5.46177 q^{27} +0.933916 q^{29} +4.70445 q^{31} +7.40473 q^{33} -7.43066 q^{35} -11.8223 q^{37} +8.09877 q^{39} +7.43611 q^{41} +7.15408 q^{43} +2.78228 q^{45} -11.7119 q^{47} +10.6800 q^{49} -5.13330 q^{51} +4.02506 q^{53} -10.9597 q^{55} +3.56281 q^{57} -9.10841 q^{59} +3.77285 q^{61} -6.61996 q^{63} -11.9869 q^{65} +9.44242 q^{67} +5.03055 q^{69} +3.31936 q^{71} -6.72218 q^{73} -2.24110 q^{75} +26.0766 q^{77} -1.40584 q^{79} -1.79808 q^{81} -13.7836 q^{83} +7.59773 q^{85} +1.11508 q^{87} +6.83991 q^{89} +28.5207 q^{91} +5.61704 q^{93} -5.27327 q^{95} -16.1719 q^{97} -9.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10}) \) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.19399 0.689348 0.344674 0.938722i \(-0.387989\pi\)
0.344674 + 0.938722i \(0.387989\pi\)
\(4\) 0 0
\(5\) −1.76720 −0.790318 −0.395159 0.918613i \(-0.629310\pi\)
−0.395159 + 0.918613i \(0.629310\pi\)
\(6\) 0 0
\(7\) 4.20476 1.58925 0.794624 0.607102i \(-0.207668\pi\)
0.794624 + 0.607102i \(0.207668\pi\)
\(8\) 0 0
\(9\) −1.57440 −0.524799
\(10\) 0 0
\(11\) 6.20169 1.86988 0.934940 0.354805i \(-0.115453\pi\)
0.934940 + 0.354805i \(0.115453\pi\)
\(12\) 0 0
\(13\) 6.78297 1.88126 0.940629 0.339437i \(-0.110236\pi\)
0.940629 + 0.339437i \(0.110236\pi\)
\(14\) 0 0
\(15\) −2.11002 −0.544804
\(16\) 0 0
\(17\) −4.29929 −1.04273 −0.521366 0.853333i \(-0.674577\pi\)
−0.521366 + 0.853333i \(0.674577\pi\)
\(18\) 0 0
\(19\) 2.98396 0.684568 0.342284 0.939596i \(-0.388799\pi\)
0.342284 + 0.939596i \(0.388799\pi\)
\(20\) 0 0
\(21\) 5.02042 1.09554
\(22\) 0 0
\(23\) 4.21325 0.878522 0.439261 0.898359i \(-0.355240\pi\)
0.439261 + 0.898359i \(0.355240\pi\)
\(24\) 0 0
\(25\) −1.87699 −0.375398
\(26\) 0 0
\(27\) −5.46177 −1.05112
\(28\) 0 0
\(29\) 0.933916 0.173424 0.0867119 0.996233i \(-0.472364\pi\)
0.0867119 + 0.996233i \(0.472364\pi\)
\(30\) 0 0
\(31\) 4.70445 0.844944 0.422472 0.906376i \(-0.361163\pi\)
0.422472 + 0.906376i \(0.361163\pi\)
\(32\) 0 0
\(33\) 7.40473 1.28900
\(34\) 0 0
\(35\) −7.43066 −1.25601
\(36\) 0 0
\(37\) −11.8223 −1.94357 −0.971784 0.235872i \(-0.924205\pi\)
−0.971784 + 0.235872i \(0.924205\pi\)
\(38\) 0 0
\(39\) 8.09877 1.29684
\(40\) 0 0
\(41\) 7.43611 1.16133 0.580663 0.814144i \(-0.302793\pi\)
0.580663 + 0.814144i \(0.302793\pi\)
\(42\) 0 0
\(43\) 7.15408 1.09099 0.545494 0.838115i \(-0.316342\pi\)
0.545494 + 0.838115i \(0.316342\pi\)
\(44\) 0 0
\(45\) 2.78228 0.414758
\(46\) 0 0
\(47\) −11.7119 −1.70836 −0.854182 0.519975i \(-0.825941\pi\)
−0.854182 + 0.519975i \(0.825941\pi\)
\(48\) 0 0
\(49\) 10.6800 1.52571
\(50\) 0 0
\(51\) −5.13330 −0.718805
\(52\) 0 0
\(53\) 4.02506 0.552884 0.276442 0.961031i \(-0.410845\pi\)
0.276442 + 0.961031i \(0.410845\pi\)
\(54\) 0 0
\(55\) −10.9597 −1.47780
\(56\) 0 0
\(57\) 3.56281 0.471906
\(58\) 0 0
\(59\) −9.10841 −1.18581 −0.592907 0.805271i \(-0.702020\pi\)
−0.592907 + 0.805271i \(0.702020\pi\)
\(60\) 0 0
\(61\) 3.77285 0.483064 0.241532 0.970393i \(-0.422350\pi\)
0.241532 + 0.970393i \(0.422350\pi\)
\(62\) 0 0
\(63\) −6.61996 −0.834036
\(64\) 0 0
\(65\) −11.9869 −1.48679
\(66\) 0 0
\(67\) 9.44242 1.15358 0.576788 0.816894i \(-0.304306\pi\)
0.576788 + 0.816894i \(0.304306\pi\)
\(68\) 0 0
\(69\) 5.03055 0.605608
\(70\) 0 0
\(71\) 3.31936 0.393935 0.196968 0.980410i \(-0.436891\pi\)
0.196968 + 0.980410i \(0.436891\pi\)
\(72\) 0 0
\(73\) −6.72218 −0.786772 −0.393386 0.919373i \(-0.628696\pi\)
−0.393386 + 0.919373i \(0.628696\pi\)
\(74\) 0 0
\(75\) −2.24110 −0.258780
\(76\) 0 0
\(77\) 26.0766 2.97170
\(78\) 0 0
\(79\) −1.40584 −0.158169 −0.0790847 0.996868i \(-0.525200\pi\)
−0.0790847 + 0.996868i \(0.525200\pi\)
\(80\) 0 0
\(81\) −1.79808 −0.199786
\(82\) 0 0
\(83\) −13.7836 −1.51295 −0.756475 0.654023i \(-0.773080\pi\)
−0.756475 + 0.654023i \(0.773080\pi\)
\(84\) 0 0
\(85\) 7.59773 0.824090
\(86\) 0 0
\(87\) 1.11508 0.119549
\(88\) 0 0
\(89\) 6.83991 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(90\) 0 0
\(91\) 28.5207 2.98979
\(92\) 0 0
\(93\) 5.61704 0.582460
\(94\) 0 0
\(95\) −5.27327 −0.541027
\(96\) 0 0
\(97\) −16.1719 −1.64200 −0.821001 0.570926i \(-0.806584\pi\)
−0.821001 + 0.570926i \(0.806584\pi\)
\(98\) 0 0
\(99\) −9.76393 −0.981312
\(100\) 0 0
\(101\) −3.58558 −0.356778 −0.178389 0.983960i \(-0.557089\pi\)
−0.178389 + 0.983960i \(0.557089\pi\)
\(102\) 0 0
\(103\) −14.3350 −1.41247 −0.706234 0.707979i \(-0.749607\pi\)
−0.706234 + 0.707979i \(0.749607\pi\)
\(104\) 0 0
\(105\) −8.87211 −0.865829
\(106\) 0 0
\(107\) 3.35920 0.324746 0.162373 0.986729i \(-0.448085\pi\)
0.162373 + 0.986729i \(0.448085\pi\)
\(108\) 0 0
\(109\) 1.50830 0.144469 0.0722343 0.997388i \(-0.476987\pi\)
0.0722343 + 0.997388i \(0.476987\pi\)
\(110\) 0 0
\(111\) −14.1156 −1.33979
\(112\) 0 0
\(113\) −2.50566 −0.235712 −0.117856 0.993031i \(-0.537602\pi\)
−0.117856 + 0.993031i \(0.537602\pi\)
\(114\) 0 0
\(115\) −7.44567 −0.694312
\(116\) 0 0
\(117\) −10.6791 −0.987283
\(118\) 0 0
\(119\) −18.0775 −1.65716
\(120\) 0 0
\(121\) 27.4610 2.49645
\(122\) 0 0
\(123\) 8.87861 0.800558
\(124\) 0 0
\(125\) 12.1530 1.08700
\(126\) 0 0
\(127\) 5.20602 0.461959 0.230980 0.972959i \(-0.425807\pi\)
0.230980 + 0.972959i \(0.425807\pi\)
\(128\) 0 0
\(129\) 8.54187 0.752070
\(130\) 0 0
\(131\) 3.65693 0.319507 0.159754 0.987157i \(-0.448930\pi\)
0.159754 + 0.987157i \(0.448930\pi\)
\(132\) 0 0
\(133\) 12.5468 1.08795
\(134\) 0 0
\(135\) 9.65206 0.830717
\(136\) 0 0
\(137\) 0.705557 0.0602798 0.0301399 0.999546i \(-0.490405\pi\)
0.0301399 + 0.999546i \(0.490405\pi\)
\(138\) 0 0
\(139\) 17.8185 1.51135 0.755675 0.654947i \(-0.227309\pi\)
0.755675 + 0.654947i \(0.227309\pi\)
\(140\) 0 0
\(141\) −13.9839 −1.17766
\(142\) 0 0
\(143\) 42.0659 3.51773
\(144\) 0 0
\(145\) −1.65042 −0.137060
\(146\) 0 0
\(147\) 12.7517 1.05174
\(148\) 0 0
\(149\) −16.4822 −1.35027 −0.675136 0.737693i \(-0.735915\pi\)
−0.675136 + 0.737693i \(0.735915\pi\)
\(150\) 0 0
\(151\) 21.4506 1.74563 0.872814 0.488054i \(-0.162293\pi\)
0.872814 + 0.488054i \(0.162293\pi\)
\(152\) 0 0
\(153\) 6.76880 0.547225
\(154\) 0 0
\(155\) −8.31372 −0.667774
\(156\) 0 0
\(157\) −11.5350 −0.920595 −0.460297 0.887765i \(-0.652257\pi\)
−0.460297 + 0.887765i \(0.652257\pi\)
\(158\) 0 0
\(159\) 4.80586 0.381129
\(160\) 0 0
\(161\) 17.7157 1.39619
\(162\) 0 0
\(163\) −1.44387 −0.113092 −0.0565461 0.998400i \(-0.518009\pi\)
−0.0565461 + 0.998400i \(0.518009\pi\)
\(164\) 0 0
\(165\) −13.0857 −1.01872
\(166\) 0 0
\(167\) −21.5365 −1.66654 −0.833271 0.552864i \(-0.813535\pi\)
−0.833271 + 0.552864i \(0.813535\pi\)
\(168\) 0 0
\(169\) 33.0087 2.53913
\(170\) 0 0
\(171\) −4.69795 −0.359261
\(172\) 0 0
\(173\) 11.6116 0.882815 0.441407 0.897307i \(-0.354479\pi\)
0.441407 + 0.897307i \(0.354479\pi\)
\(174\) 0 0
\(175\) −7.89227 −0.596600
\(176\) 0 0
\(177\) −10.8753 −0.817438
\(178\) 0 0
\(179\) 4.67436 0.349378 0.174689 0.984624i \(-0.444108\pi\)
0.174689 + 0.984624i \(0.444108\pi\)
\(180\) 0 0
\(181\) −10.7227 −0.797010 −0.398505 0.917166i \(-0.630471\pi\)
−0.398505 + 0.917166i \(0.630471\pi\)
\(182\) 0 0
\(183\) 4.50473 0.332999
\(184\) 0 0
\(185\) 20.8924 1.53604
\(186\) 0 0
\(187\) −26.6629 −1.94978
\(188\) 0 0
\(189\) −22.9654 −1.67049
\(190\) 0 0
\(191\) −7.67697 −0.555486 −0.277743 0.960655i \(-0.589586\pi\)
−0.277743 + 0.960655i \(0.589586\pi\)
\(192\) 0 0
\(193\) 19.4524 1.40021 0.700106 0.714039i \(-0.253136\pi\)
0.700106 + 0.714039i \(0.253136\pi\)
\(194\) 0 0
\(195\) −14.3122 −1.02492
\(196\) 0 0
\(197\) −0.619732 −0.0441541 −0.0220770 0.999756i \(-0.507028\pi\)
−0.0220770 + 0.999756i \(0.507028\pi\)
\(198\) 0 0
\(199\) 11.4611 0.812453 0.406226 0.913772i \(-0.366844\pi\)
0.406226 + 0.913772i \(0.366844\pi\)
\(200\) 0 0
\(201\) 11.2741 0.795215
\(202\) 0 0
\(203\) 3.92689 0.275614
\(204\) 0 0
\(205\) −13.1411 −0.917817
\(206\) 0 0
\(207\) −6.63333 −0.461048
\(208\) 0 0
\(209\) 18.5056 1.28006
\(210\) 0 0
\(211\) 24.2699 1.67081 0.835405 0.549634i \(-0.185233\pi\)
0.835405 + 0.549634i \(0.185233\pi\)
\(212\) 0 0
\(213\) 3.96327 0.271559
\(214\) 0 0
\(215\) −12.6427 −0.862227
\(216\) 0 0
\(217\) 19.7811 1.34283
\(218\) 0 0
\(219\) −8.02619 −0.542360
\(220\) 0 0
\(221\) −29.1620 −1.96165
\(222\) 0 0
\(223\) −11.8007 −0.790234 −0.395117 0.918631i \(-0.629296\pi\)
−0.395117 + 0.918631i \(0.629296\pi\)
\(224\) 0 0
\(225\) 2.95513 0.197008
\(226\) 0 0
\(227\) 12.6700 0.840939 0.420470 0.907307i \(-0.361865\pi\)
0.420470 + 0.907307i \(0.361865\pi\)
\(228\) 0 0
\(229\) 17.3224 1.14469 0.572347 0.820012i \(-0.306033\pi\)
0.572347 + 0.820012i \(0.306033\pi\)
\(230\) 0 0
\(231\) 31.1351 2.04854
\(232\) 0 0
\(233\) 10.9692 0.718614 0.359307 0.933219i \(-0.383013\pi\)
0.359307 + 0.933219i \(0.383013\pi\)
\(234\) 0 0
\(235\) 20.6974 1.35015
\(236\) 0 0
\(237\) −1.67855 −0.109034
\(238\) 0 0
\(239\) −3.70714 −0.239795 −0.119898 0.992786i \(-0.538257\pi\)
−0.119898 + 0.992786i \(0.538257\pi\)
\(240\) 0 0
\(241\) 23.2511 1.49773 0.748866 0.662721i \(-0.230599\pi\)
0.748866 + 0.662721i \(0.230599\pi\)
\(242\) 0 0
\(243\) 14.2384 0.913395
\(244\) 0 0
\(245\) −18.8737 −1.20580
\(246\) 0 0
\(247\) 20.2401 1.28785
\(248\) 0 0
\(249\) −16.4575 −1.04295
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 26.1293 1.64273
\(254\) 0 0
\(255\) 9.07159 0.568085
\(256\) 0 0
\(257\) −11.6859 −0.728945 −0.364473 0.931214i \(-0.618751\pi\)
−0.364473 + 0.931214i \(0.618751\pi\)
\(258\) 0 0
\(259\) −49.7097 −3.08881
\(260\) 0 0
\(261\) −1.47036 −0.0910127
\(262\) 0 0
\(263\) −3.22217 −0.198687 −0.0993437 0.995053i \(-0.531674\pi\)
−0.0993437 + 0.995053i \(0.531674\pi\)
\(264\) 0 0
\(265\) −7.11310 −0.436954
\(266\) 0 0
\(267\) 8.16676 0.499798
\(268\) 0 0
\(269\) −21.4922 −1.31041 −0.655203 0.755453i \(-0.727417\pi\)
−0.655203 + 0.755453i \(0.727417\pi\)
\(270\) 0 0
\(271\) 17.1881 1.04410 0.522050 0.852915i \(-0.325167\pi\)
0.522050 + 0.852915i \(0.325167\pi\)
\(272\) 0 0
\(273\) 34.0534 2.06100
\(274\) 0 0
\(275\) −11.6405 −0.701949
\(276\) 0 0
\(277\) −14.3703 −0.863431 −0.431715 0.902010i \(-0.642091\pi\)
−0.431715 + 0.902010i \(0.642091\pi\)
\(278\) 0 0
\(279\) −7.40667 −0.443426
\(280\) 0 0
\(281\) 17.7574 1.05932 0.529659 0.848211i \(-0.322320\pi\)
0.529659 + 0.848211i \(0.322320\pi\)
\(282\) 0 0
\(283\) −24.9012 −1.48023 −0.740113 0.672483i \(-0.765228\pi\)
−0.740113 + 0.672483i \(0.765228\pi\)
\(284\) 0 0
\(285\) −6.29621 −0.372956
\(286\) 0 0
\(287\) 31.2670 1.84564
\(288\) 0 0
\(289\) 1.48394 0.0872903
\(290\) 0 0
\(291\) −19.3090 −1.13191
\(292\) 0 0
\(293\) −26.4442 −1.54489 −0.772444 0.635083i \(-0.780966\pi\)
−0.772444 + 0.635083i \(0.780966\pi\)
\(294\) 0 0
\(295\) 16.0964 0.937170
\(296\) 0 0
\(297\) −33.8722 −1.96546
\(298\) 0 0
\(299\) 28.5783 1.65273
\(300\) 0 0
\(301\) 30.0812 1.73385
\(302\) 0 0
\(303\) −4.28113 −0.245944
\(304\) 0 0
\(305\) −6.66740 −0.381774
\(306\) 0 0
\(307\) −2.58925 −0.147776 −0.0738881 0.997267i \(-0.523541\pi\)
−0.0738881 + 0.997267i \(0.523541\pi\)
\(308\) 0 0
\(309\) −17.1158 −0.973682
\(310\) 0 0
\(311\) 33.6678 1.90912 0.954562 0.298012i \(-0.0963234\pi\)
0.954562 + 0.298012i \(0.0963234\pi\)
\(312\) 0 0
\(313\) 2.93749 0.166037 0.0830184 0.996548i \(-0.473544\pi\)
0.0830184 + 0.996548i \(0.473544\pi\)
\(314\) 0 0
\(315\) 11.6988 0.659154
\(316\) 0 0
\(317\) −3.49082 −0.196064 −0.0980320 0.995183i \(-0.531255\pi\)
−0.0980320 + 0.995183i \(0.531255\pi\)
\(318\) 0 0
\(319\) 5.79186 0.324282
\(320\) 0 0
\(321\) 4.01083 0.223863
\(322\) 0 0
\(323\) −12.8289 −0.713821
\(324\) 0 0
\(325\) −12.7316 −0.706220
\(326\) 0 0
\(327\) 1.80088 0.0995891
\(328\) 0 0
\(329\) −49.2459 −2.71501
\(330\) 0 0
\(331\) −12.8901 −0.708502 −0.354251 0.935150i \(-0.615264\pi\)
−0.354251 + 0.935150i \(0.615264\pi\)
\(332\) 0 0
\(333\) 18.6129 1.01998
\(334\) 0 0
\(335\) −16.6867 −0.911691
\(336\) 0 0
\(337\) −8.10711 −0.441622 −0.220811 0.975317i \(-0.570870\pi\)
−0.220811 + 0.975317i \(0.570870\pi\)
\(338\) 0 0
\(339\) −2.99172 −0.162488
\(340\) 0 0
\(341\) 29.1755 1.57994
\(342\) 0 0
\(343\) 15.4734 0.835483
\(344\) 0 0
\(345\) −8.89002 −0.478623
\(346\) 0 0
\(347\) 0.219765 0.0117976 0.00589879 0.999983i \(-0.498122\pi\)
0.00589879 + 0.999983i \(0.498122\pi\)
\(348\) 0 0
\(349\) −6.61716 −0.354208 −0.177104 0.984192i \(-0.556673\pi\)
−0.177104 + 0.984192i \(0.556673\pi\)
\(350\) 0 0
\(351\) −37.0470 −1.97742
\(352\) 0 0
\(353\) −5.86366 −0.312091 −0.156046 0.987750i \(-0.549875\pi\)
−0.156046 + 0.987750i \(0.549875\pi\)
\(354\) 0 0
\(355\) −5.86599 −0.311334
\(356\) 0 0
\(357\) −21.5843 −1.14236
\(358\) 0 0
\(359\) −13.2624 −0.699963 −0.349981 0.936757i \(-0.613812\pi\)
−0.349981 + 0.936757i \(0.613812\pi\)
\(360\) 0 0
\(361\) −10.0960 −0.531366
\(362\) 0 0
\(363\) 32.7880 1.72093
\(364\) 0 0
\(365\) 11.8795 0.621800
\(366\) 0 0
\(367\) 4.78717 0.249888 0.124944 0.992164i \(-0.460125\pi\)
0.124944 + 0.992164i \(0.460125\pi\)
\(368\) 0 0
\(369\) −11.7074 −0.609463
\(370\) 0 0
\(371\) 16.9244 0.878670
\(372\) 0 0
\(373\) −24.2527 −1.25576 −0.627879 0.778311i \(-0.716077\pi\)
−0.627879 + 0.778311i \(0.716077\pi\)
\(374\) 0 0
\(375\) 14.5106 0.749322
\(376\) 0 0
\(377\) 6.33473 0.326255
\(378\) 0 0
\(379\) −13.2484 −0.680527 −0.340263 0.940330i \(-0.610516\pi\)
−0.340263 + 0.940330i \(0.610516\pi\)
\(380\) 0 0
\(381\) 6.21591 0.318451
\(382\) 0 0
\(383\) 5.47967 0.279998 0.139999 0.990152i \(-0.455290\pi\)
0.139999 + 0.990152i \(0.455290\pi\)
\(384\) 0 0
\(385\) −46.0827 −2.34859
\(386\) 0 0
\(387\) −11.2634 −0.572550
\(388\) 0 0
\(389\) −34.2471 −1.73640 −0.868200 0.496215i \(-0.834723\pi\)
−0.868200 + 0.496215i \(0.834723\pi\)
\(390\) 0 0
\(391\) −18.1140 −0.916063
\(392\) 0 0
\(393\) 4.36632 0.220252
\(394\) 0 0
\(395\) 2.48441 0.125004
\(396\) 0 0
\(397\) 7.08290 0.355481 0.177740 0.984077i \(-0.443121\pi\)
0.177740 + 0.984077i \(0.443121\pi\)
\(398\) 0 0
\(399\) 14.9807 0.749975
\(400\) 0 0
\(401\) −7.12305 −0.355708 −0.177854 0.984057i \(-0.556915\pi\)
−0.177854 + 0.984057i \(0.556915\pi\)
\(402\) 0 0
\(403\) 31.9101 1.58956
\(404\) 0 0
\(405\) 3.17757 0.157895
\(406\) 0 0
\(407\) −73.3180 −3.63424
\(408\) 0 0
\(409\) 19.1924 0.949001 0.474500 0.880255i \(-0.342629\pi\)
0.474500 + 0.880255i \(0.342629\pi\)
\(410\) 0 0
\(411\) 0.842424 0.0415537
\(412\) 0 0
\(413\) −38.2986 −1.88455
\(414\) 0 0
\(415\) 24.3585 1.19571
\(416\) 0 0
\(417\) 21.2751 1.04185
\(418\) 0 0
\(419\) 2.74077 0.133896 0.0669478 0.997756i \(-0.478674\pi\)
0.0669478 + 0.997756i \(0.478674\pi\)
\(420\) 0 0
\(421\) −12.0522 −0.587390 −0.293695 0.955899i \(-0.594885\pi\)
−0.293695 + 0.955899i \(0.594885\pi\)
\(422\) 0 0
\(423\) 18.4393 0.896548
\(424\) 0 0
\(425\) 8.06972 0.391439
\(426\) 0 0
\(427\) 15.8639 0.767709
\(428\) 0 0
\(429\) 50.2261 2.42494
\(430\) 0 0
\(431\) −41.0247 −1.97609 −0.988046 0.154157i \(-0.950734\pi\)
−0.988046 + 0.154157i \(0.950734\pi\)
\(432\) 0 0
\(433\) 9.72121 0.467171 0.233586 0.972336i \(-0.424954\pi\)
0.233586 + 0.972336i \(0.424954\pi\)
\(434\) 0 0
\(435\) −1.97058 −0.0944820
\(436\) 0 0
\(437\) 12.5722 0.601409
\(438\) 0 0
\(439\) 25.3212 1.20852 0.604259 0.796788i \(-0.293469\pi\)
0.604259 + 0.796788i \(0.293469\pi\)
\(440\) 0 0
\(441\) −16.8145 −0.800692
\(442\) 0 0
\(443\) −19.1599 −0.910314 −0.455157 0.890411i \(-0.650417\pi\)
−0.455157 + 0.890411i \(0.650417\pi\)
\(444\) 0 0
\(445\) −12.0875 −0.573004
\(446\) 0 0
\(447\) −19.6795 −0.930807
\(448\) 0 0
\(449\) 12.8326 0.605607 0.302803 0.953053i \(-0.402077\pi\)
0.302803 + 0.953053i \(0.402077\pi\)
\(450\) 0 0
\(451\) 46.1165 2.17154
\(452\) 0 0
\(453\) 25.6117 1.20334
\(454\) 0 0
\(455\) −50.4020 −2.36288
\(456\) 0 0
\(457\) 16.7454 0.783315 0.391658 0.920111i \(-0.371902\pi\)
0.391658 + 0.920111i \(0.371902\pi\)
\(458\) 0 0
\(459\) 23.4817 1.09603
\(460\) 0 0
\(461\) −30.7302 −1.43125 −0.715624 0.698486i \(-0.753858\pi\)
−0.715624 + 0.698486i \(0.753858\pi\)
\(462\) 0 0
\(463\) 16.1963 0.752706 0.376353 0.926476i \(-0.377178\pi\)
0.376353 + 0.926476i \(0.377178\pi\)
\(464\) 0 0
\(465\) −9.92646 −0.460329
\(466\) 0 0
\(467\) −5.70856 −0.264161 −0.132080 0.991239i \(-0.542166\pi\)
−0.132080 + 0.991239i \(0.542166\pi\)
\(468\) 0 0
\(469\) 39.7031 1.83332
\(470\) 0 0
\(471\) −13.7726 −0.634610
\(472\) 0 0
\(473\) 44.3674 2.04002
\(474\) 0 0
\(475\) −5.60086 −0.256985
\(476\) 0 0
\(477\) −6.33704 −0.290153
\(478\) 0 0
\(479\) −31.0622 −1.41927 −0.709634 0.704570i \(-0.751140\pi\)
−0.709634 + 0.704570i \(0.751140\pi\)
\(480\) 0 0
\(481\) −80.1901 −3.65635
\(482\) 0 0
\(483\) 21.1523 0.962461
\(484\) 0 0
\(485\) 28.5790 1.29770
\(486\) 0 0
\(487\) 3.41545 0.154769 0.0773844 0.997001i \(-0.475343\pi\)
0.0773844 + 0.997001i \(0.475343\pi\)
\(488\) 0 0
\(489\) −1.72395 −0.0779599
\(490\) 0 0
\(491\) 7.92993 0.357873 0.178936 0.983861i \(-0.442734\pi\)
0.178936 + 0.983861i \(0.442734\pi\)
\(492\) 0 0
\(493\) −4.01518 −0.180835
\(494\) 0 0
\(495\) 17.2549 0.775549
\(496\) 0 0
\(497\) 13.9571 0.626061
\(498\) 0 0
\(499\) −3.80951 −0.170537 −0.0852685 0.996358i \(-0.527175\pi\)
−0.0852685 + 0.996358i \(0.527175\pi\)
\(500\) 0 0
\(501\) −25.7142 −1.14883
\(502\) 0 0
\(503\) 15.1279 0.674518 0.337259 0.941412i \(-0.390500\pi\)
0.337259 + 0.941412i \(0.390500\pi\)
\(504\) 0 0
\(505\) 6.33645 0.281968
\(506\) 0 0
\(507\) 39.4119 1.75034
\(508\) 0 0
\(509\) −6.07898 −0.269446 −0.134723 0.990883i \(-0.543014\pi\)
−0.134723 + 0.990883i \(0.543014\pi\)
\(510\) 0 0
\(511\) −28.2651 −1.25038
\(512\) 0 0
\(513\) −16.2977 −0.719562
\(514\) 0 0
\(515\) 25.3328 1.11630
\(516\) 0 0
\(517\) −72.6339 −3.19444
\(518\) 0 0
\(519\) 13.8641 0.608567
\(520\) 0 0
\(521\) −5.12024 −0.224322 −0.112161 0.993690i \(-0.535777\pi\)
−0.112161 + 0.993690i \(0.535777\pi\)
\(522\) 0 0
\(523\) −16.5817 −0.725068 −0.362534 0.931971i \(-0.618088\pi\)
−0.362534 + 0.931971i \(0.618088\pi\)
\(524\) 0 0
\(525\) −9.42326 −0.411265
\(526\) 0 0
\(527\) −20.2258 −0.881050
\(528\) 0 0
\(529\) −5.24857 −0.228199
\(530\) 0 0
\(531\) 14.3403 0.622314
\(532\) 0 0
\(533\) 50.4390 2.18475
\(534\) 0 0
\(535\) −5.93639 −0.256652
\(536\) 0 0
\(537\) 5.58111 0.240843
\(538\) 0 0
\(539\) 66.2339 2.85290
\(540\) 0 0
\(541\) −21.4638 −0.922800 −0.461400 0.887192i \(-0.652653\pi\)
−0.461400 + 0.887192i \(0.652653\pi\)
\(542\) 0 0
\(543\) −12.8027 −0.549417
\(544\) 0 0
\(545\) −2.66547 −0.114176
\(546\) 0 0
\(547\) −40.6068 −1.73622 −0.868110 0.496372i \(-0.834665\pi\)
−0.868110 + 0.496372i \(0.834665\pi\)
\(548\) 0 0
\(549\) −5.93997 −0.253512
\(550\) 0 0
\(551\) 2.78677 0.118720
\(552\) 0 0
\(553\) −5.91122 −0.251370
\(554\) 0 0
\(555\) 24.9452 1.05886
\(556\) 0 0
\(557\) 40.3887 1.71132 0.855662 0.517534i \(-0.173150\pi\)
0.855662 + 0.517534i \(0.173150\pi\)
\(558\) 0 0
\(559\) 48.5259 2.05243
\(560\) 0 0
\(561\) −31.8351 −1.34408
\(562\) 0 0
\(563\) −17.6645 −0.744468 −0.372234 0.928139i \(-0.621408\pi\)
−0.372234 + 0.928139i \(0.621408\pi\)
\(564\) 0 0
\(565\) 4.42801 0.186288
\(566\) 0 0
\(567\) −7.56047 −0.317510
\(568\) 0 0
\(569\) −8.47503 −0.355292 −0.177646 0.984094i \(-0.556848\pi\)
−0.177646 + 0.984094i \(0.556848\pi\)
\(570\) 0 0
\(571\) −22.2737 −0.932126 −0.466063 0.884751i \(-0.654328\pi\)
−0.466063 + 0.884751i \(0.654328\pi\)
\(572\) 0 0
\(573\) −9.16619 −0.382923
\(574\) 0 0
\(575\) −7.90821 −0.329795
\(576\) 0 0
\(577\) 30.7794 1.28137 0.640683 0.767806i \(-0.278651\pi\)
0.640683 + 0.767806i \(0.278651\pi\)
\(578\) 0 0
\(579\) 23.2258 0.965233
\(580\) 0 0
\(581\) −57.9568 −2.40445
\(582\) 0 0
\(583\) 24.9622 1.03383
\(584\) 0 0
\(585\) 18.8722 0.780267
\(586\) 0 0
\(587\) 7.20381 0.297333 0.148667 0.988887i \(-0.452502\pi\)
0.148667 + 0.988887i \(0.452502\pi\)
\(588\) 0 0
\(589\) 14.0379 0.578422
\(590\) 0 0
\(591\) −0.739951 −0.0304375
\(592\) 0 0
\(593\) 16.0481 0.659015 0.329508 0.944153i \(-0.393117\pi\)
0.329508 + 0.944153i \(0.393117\pi\)
\(594\) 0 0
\(595\) 31.9466 1.30968
\(596\) 0 0
\(597\) 13.6843 0.560063
\(598\) 0 0
\(599\) −5.40921 −0.221014 −0.110507 0.993875i \(-0.535248\pi\)
−0.110507 + 0.993875i \(0.535248\pi\)
\(600\) 0 0
\(601\) 12.6124 0.514472 0.257236 0.966349i \(-0.417188\pi\)
0.257236 + 0.966349i \(0.417188\pi\)
\(602\) 0 0
\(603\) −14.8661 −0.605396
\(604\) 0 0
\(605\) −48.5292 −1.97299
\(606\) 0 0
\(607\) −27.0031 −1.09602 −0.548012 0.836470i \(-0.684615\pi\)
−0.548012 + 0.836470i \(0.684615\pi\)
\(608\) 0 0
\(609\) 4.68865 0.189994
\(610\) 0 0
\(611\) −79.4418 −3.21387
\(612\) 0 0
\(613\) −28.9168 −1.16794 −0.583970 0.811775i \(-0.698502\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(614\) 0 0
\(615\) −15.6903 −0.632695
\(616\) 0 0
\(617\) 22.5144 0.906397 0.453199 0.891410i \(-0.350283\pi\)
0.453199 + 0.891410i \(0.350283\pi\)
\(618\) 0 0
\(619\) 29.5038 1.18586 0.592929 0.805255i \(-0.297971\pi\)
0.592929 + 0.805255i \(0.297971\pi\)
\(620\) 0 0
\(621\) −23.0118 −0.923430
\(622\) 0 0
\(623\) 28.7602 1.15225
\(624\) 0 0
\(625\) −12.0920 −0.483679
\(626\) 0 0
\(627\) 22.0955 0.882407
\(628\) 0 0
\(629\) 50.8274 2.02662
\(630\) 0 0
\(631\) 28.3742 1.12956 0.564779 0.825242i \(-0.308961\pi\)
0.564779 + 0.825242i \(0.308961\pi\)
\(632\) 0 0
\(633\) 28.9779 1.15177
\(634\) 0 0
\(635\) −9.20010 −0.365095
\(636\) 0 0
\(637\) 72.4419 2.87025
\(638\) 0 0
\(639\) −5.22599 −0.206737
\(640\) 0 0
\(641\) −32.7878 −1.29504 −0.647521 0.762048i \(-0.724194\pi\)
−0.647521 + 0.762048i \(0.724194\pi\)
\(642\) 0 0
\(643\) −23.7781 −0.937715 −0.468857 0.883274i \(-0.655334\pi\)
−0.468857 + 0.883274i \(0.655334\pi\)
\(644\) 0 0
\(645\) −15.0952 −0.594374
\(646\) 0 0
\(647\) −4.53047 −0.178111 −0.0890555 0.996027i \(-0.528385\pi\)
−0.0890555 + 0.996027i \(0.528385\pi\)
\(648\) 0 0
\(649\) −56.4876 −2.21733
\(650\) 0 0
\(651\) 23.6183 0.925674
\(652\) 0 0
\(653\) −2.54926 −0.0997604 −0.0498802 0.998755i \(-0.515884\pi\)
−0.0498802 + 0.998755i \(0.515884\pi\)
\(654\) 0 0
\(655\) −6.46254 −0.252513
\(656\) 0 0
\(657\) 10.5834 0.412898
\(658\) 0 0
\(659\) −13.4708 −0.524748 −0.262374 0.964966i \(-0.584505\pi\)
−0.262374 + 0.964966i \(0.584505\pi\)
\(660\) 0 0
\(661\) 34.6527 1.34783 0.673917 0.738807i \(-0.264611\pi\)
0.673917 + 0.738807i \(0.264611\pi\)
\(662\) 0 0
\(663\) −34.8190 −1.35226
\(664\) 0 0
\(665\) −22.1728 −0.859826
\(666\) 0 0
\(667\) 3.93482 0.152357
\(668\) 0 0
\(669\) −14.0899 −0.544746
\(670\) 0 0
\(671\) 23.3981 0.903272
\(672\) 0 0
\(673\) −1.08294 −0.0417441 −0.0208720 0.999782i \(-0.506644\pi\)
−0.0208720 + 0.999782i \(0.506644\pi\)
\(674\) 0 0
\(675\) 10.2517 0.394587
\(676\) 0 0
\(677\) 28.4129 1.09200 0.545998 0.837787i \(-0.316151\pi\)
0.545998 + 0.837787i \(0.316151\pi\)
\(678\) 0 0
\(679\) −67.9987 −2.60955
\(680\) 0 0
\(681\) 15.1278 0.579700
\(682\) 0 0
\(683\) 6.95074 0.265963 0.132981 0.991119i \(-0.457545\pi\)
0.132981 + 0.991119i \(0.457545\pi\)
\(684\) 0 0
\(685\) −1.24686 −0.0476402
\(686\) 0 0
\(687\) 20.6827 0.789092
\(688\) 0 0
\(689\) 27.3018 1.04012
\(690\) 0 0
\(691\) −47.6966 −1.81446 −0.907232 0.420630i \(-0.861809\pi\)
−0.907232 + 0.420630i \(0.861809\pi\)
\(692\) 0 0
\(693\) −41.0550 −1.55955
\(694\) 0 0
\(695\) −31.4890 −1.19445
\(696\) 0 0
\(697\) −31.9701 −1.21095
\(698\) 0 0
\(699\) 13.0970 0.495375
\(700\) 0 0
\(701\) −40.9673 −1.54731 −0.773657 0.633604i \(-0.781575\pi\)
−0.773657 + 0.633604i \(0.781575\pi\)
\(702\) 0 0
\(703\) −35.2772 −1.33051
\(704\) 0 0
\(705\) 24.7124 0.930723
\(706\) 0 0
\(707\) −15.0765 −0.567009
\(708\) 0 0
\(709\) 7.52985 0.282790 0.141395 0.989953i \(-0.454841\pi\)
0.141395 + 0.989953i \(0.454841\pi\)
\(710\) 0 0
\(711\) 2.21335 0.0830072
\(712\) 0 0
\(713\) 19.8210 0.742302
\(714\) 0 0
\(715\) −74.3391 −2.78012
\(716\) 0 0
\(717\) −4.42628 −0.165302
\(718\) 0 0
\(719\) −4.52850 −0.168885 −0.0844423 0.996428i \(-0.526911\pi\)
−0.0844423 + 0.996428i \(0.526911\pi\)
\(720\) 0 0
\(721\) −60.2751 −2.24476
\(722\) 0 0
\(723\) 27.7614 1.03246
\(724\) 0 0
\(725\) −1.75295 −0.0651029
\(726\) 0 0
\(727\) −5.37432 −0.199322 −0.0996612 0.995021i \(-0.531776\pi\)
−0.0996612 + 0.995021i \(0.531776\pi\)
\(728\) 0 0
\(729\) 22.3947 0.829433
\(730\) 0 0
\(731\) −30.7575 −1.13761
\(732\) 0 0
\(733\) −30.7891 −1.13722 −0.568610 0.822607i \(-0.692519\pi\)
−0.568610 + 0.822607i \(0.692519\pi\)
\(734\) 0 0
\(735\) −22.5349 −0.831213
\(736\) 0 0
\(737\) 58.5590 2.15705
\(738\) 0 0
\(739\) 16.6744 0.613378 0.306689 0.951810i \(-0.400779\pi\)
0.306689 + 0.951810i \(0.400779\pi\)
\(740\) 0 0
\(741\) 24.1664 0.887776
\(742\) 0 0
\(743\) −8.53333 −0.313058 −0.156529 0.987673i \(-0.550030\pi\)
−0.156529 + 0.987673i \(0.550030\pi\)
\(744\) 0 0
\(745\) 29.1274 1.06714
\(746\) 0 0
\(747\) 21.7009 0.793995
\(748\) 0 0
\(749\) 14.1246 0.516102
\(750\) 0 0
\(751\) 17.0102 0.620709 0.310355 0.950621i \(-0.399552\pi\)
0.310355 + 0.950621i \(0.399552\pi\)
\(752\) 0 0
\(753\) 1.19399 0.0435113
\(754\) 0 0
\(755\) −37.9076 −1.37960
\(756\) 0 0
\(757\) −4.82033 −0.175198 −0.0875989 0.996156i \(-0.527919\pi\)
−0.0875989 + 0.996156i \(0.527919\pi\)
\(758\) 0 0
\(759\) 31.1980 1.13241
\(760\) 0 0
\(761\) −25.5876 −0.927549 −0.463775 0.885953i \(-0.653505\pi\)
−0.463775 + 0.885953i \(0.653505\pi\)
\(762\) 0 0
\(763\) 6.34202 0.229596
\(764\) 0 0
\(765\) −11.9619 −0.432482
\(766\) 0 0
\(767\) −61.7821 −2.23082
\(768\) 0 0
\(769\) 53.8660 1.94246 0.971230 0.238145i \(-0.0765393\pi\)
0.971230 + 0.238145i \(0.0765393\pi\)
\(770\) 0 0
\(771\) −13.9528 −0.502497
\(772\) 0 0
\(773\) 17.6962 0.636489 0.318244 0.948009i \(-0.396907\pi\)
0.318244 + 0.948009i \(0.396907\pi\)
\(774\) 0 0
\(775\) −8.83019 −0.317190
\(776\) 0 0
\(777\) −59.3527 −2.12927
\(778\) 0 0
\(779\) 22.1891 0.795007
\(780\) 0 0
\(781\) 20.5856 0.736612
\(782\) 0 0
\(783\) −5.10083 −0.182289
\(784\) 0 0
\(785\) 20.3847 0.727562
\(786\) 0 0
\(787\) −5.44792 −0.194197 −0.0970987 0.995275i \(-0.530956\pi\)
−0.0970987 + 0.995275i \(0.530956\pi\)
\(788\) 0 0
\(789\) −3.84722 −0.136965
\(790\) 0 0
\(791\) −10.5357 −0.374606
\(792\) 0 0
\(793\) 25.5911 0.908768
\(794\) 0 0
\(795\) −8.49294 −0.301213
\(796\) 0 0
\(797\) 16.3912 0.580607 0.290304 0.956935i \(-0.406244\pi\)
0.290304 + 0.956935i \(0.406244\pi\)
\(798\) 0 0
\(799\) 50.3531 1.78137
\(800\) 0 0
\(801\) −10.7687 −0.380495
\(802\) 0 0
\(803\) −41.6889 −1.47117
\(804\) 0 0
\(805\) −31.3072 −1.10343
\(806\) 0 0
\(807\) −25.6614 −0.903325
\(808\) 0 0
\(809\) −2.33351 −0.0820417 −0.0410209 0.999158i \(-0.513061\pi\)
−0.0410209 + 0.999158i \(0.513061\pi\)
\(810\) 0 0
\(811\) −12.4035 −0.435544 −0.217772 0.976000i \(-0.569879\pi\)
−0.217772 + 0.976000i \(0.569879\pi\)
\(812\) 0 0
\(813\) 20.5223 0.719748
\(814\) 0 0
\(815\) 2.55161 0.0893788
\(816\) 0 0
\(817\) 21.3475 0.746855
\(818\) 0 0
\(819\) −44.9030 −1.56904
\(820\) 0 0
\(821\) −19.7162 −0.688101 −0.344050 0.938951i \(-0.611799\pi\)
−0.344050 + 0.938951i \(0.611799\pi\)
\(822\) 0 0
\(823\) 13.4615 0.469237 0.234619 0.972087i \(-0.424616\pi\)
0.234619 + 0.972087i \(0.424616\pi\)
\(824\) 0 0
\(825\) −13.8986 −0.483887
\(826\) 0 0
\(827\) 11.9796 0.416571 0.208286 0.978068i \(-0.433212\pi\)
0.208286 + 0.978068i \(0.433212\pi\)
\(828\) 0 0
\(829\) −8.40159 −0.291799 −0.145900 0.989299i \(-0.546608\pi\)
−0.145900 + 0.989299i \(0.546608\pi\)
\(830\) 0 0
\(831\) −17.1580 −0.595204
\(832\) 0 0
\(833\) −45.9163 −1.59091
\(834\) 0 0
\(835\) 38.0594 1.31710
\(836\) 0 0
\(837\) −25.6946 −0.888135
\(838\) 0 0
\(839\) −25.7588 −0.889291 −0.444646 0.895707i \(-0.646670\pi\)
−0.444646 + 0.895707i \(0.646670\pi\)
\(840\) 0 0
\(841\) −28.1278 −0.969924
\(842\) 0 0
\(843\) 21.2021 0.730239
\(844\) 0 0
\(845\) −58.3331 −2.00672
\(846\) 0 0
\(847\) 115.467 3.96748
\(848\) 0 0
\(849\) −29.7317 −1.02039
\(850\) 0 0
\(851\) −49.8101 −1.70747
\(852\) 0 0
\(853\) −13.6635 −0.467828 −0.233914 0.972257i \(-0.575153\pi\)
−0.233914 + 0.972257i \(0.575153\pi\)
\(854\) 0 0
\(855\) 8.30223 0.283930
\(856\) 0 0
\(857\) 34.2063 1.16846 0.584232 0.811587i \(-0.301396\pi\)
0.584232 + 0.811587i \(0.301396\pi\)
\(858\) 0 0
\(859\) −20.3749 −0.695183 −0.347592 0.937646i \(-0.613000\pi\)
−0.347592 + 0.937646i \(0.613000\pi\)
\(860\) 0 0
\(861\) 37.3324 1.27228
\(862\) 0 0
\(863\) 1.86805 0.0635890 0.0317945 0.999494i \(-0.489878\pi\)
0.0317945 + 0.999494i \(0.489878\pi\)
\(864\) 0 0
\(865\) −20.5201 −0.697705
\(866\) 0 0
\(867\) 1.77180 0.0601734
\(868\) 0 0
\(869\) −8.71859 −0.295758
\(870\) 0 0
\(871\) 64.0477 2.17017
\(872\) 0 0
\(873\) 25.4609 0.861722
\(874\) 0 0
\(875\) 51.1006 1.72751
\(876\) 0 0
\(877\) 38.6331 1.30455 0.652274 0.757984i \(-0.273815\pi\)
0.652274 + 0.757984i \(0.273815\pi\)
\(878\) 0 0
\(879\) −31.5740 −1.06497
\(880\) 0 0
\(881\) 19.5064 0.657188 0.328594 0.944471i \(-0.393425\pi\)
0.328594 + 0.944471i \(0.393425\pi\)
\(882\) 0 0
\(883\) 31.7268 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(884\) 0 0
\(885\) 19.2189 0.646036
\(886\) 0 0
\(887\) −14.5921 −0.489953 −0.244977 0.969529i \(-0.578780\pi\)
−0.244977 + 0.969529i \(0.578780\pi\)
\(888\) 0 0
\(889\) 21.8900 0.734168
\(890\) 0 0
\(891\) −11.1511 −0.373576
\(892\) 0 0
\(893\) −34.9480 −1.16949
\(894\) 0 0
\(895\) −8.26054 −0.276120
\(896\) 0 0
\(897\) 34.1221 1.13930
\(898\) 0 0
\(899\) 4.39356 0.146533
\(900\) 0 0
\(901\) −17.3049 −0.576510
\(902\) 0 0
\(903\) 35.9165 1.19523
\(904\) 0 0
\(905\) 18.9492 0.629891
\(906\) 0 0
\(907\) −57.9743 −1.92501 −0.962503 0.271271i \(-0.912556\pi\)
−0.962503 + 0.271271i \(0.912556\pi\)
\(908\) 0 0
\(909\) 5.64513 0.187237
\(910\) 0 0
\(911\) 48.3160 1.60078 0.800390 0.599479i \(-0.204626\pi\)
0.800390 + 0.599479i \(0.204626\pi\)
\(912\) 0 0
\(913\) −85.4818 −2.82904
\(914\) 0 0
\(915\) −7.96078 −0.263175
\(916\) 0 0
\(917\) 15.3765 0.507777
\(918\) 0 0
\(919\) −31.6800 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(920\) 0 0
\(921\) −3.09153 −0.101869
\(922\) 0 0
\(923\) 22.5151 0.741094
\(924\) 0 0
\(925\) 22.1902 0.729611
\(926\) 0 0
\(927\) 22.5690 0.741262
\(928\) 0 0
\(929\) 0.260768 0.00855554 0.00427777 0.999991i \(-0.498638\pi\)
0.00427777 + 0.999991i \(0.498638\pi\)
\(930\) 0 0
\(931\) 31.8686 1.04445
\(932\) 0 0
\(933\) 40.1988 1.31605
\(934\) 0 0
\(935\) 47.1188 1.54095
\(936\) 0 0
\(937\) −10.4191 −0.340377 −0.170189 0.985412i \(-0.554438\pi\)
−0.170189 + 0.985412i \(0.554438\pi\)
\(938\) 0 0
\(939\) 3.50732 0.114457
\(940\) 0 0
\(941\) 17.3927 0.566985 0.283492 0.958974i \(-0.408507\pi\)
0.283492 + 0.958974i \(0.408507\pi\)
\(942\) 0 0
\(943\) 31.3302 1.02025
\(944\) 0 0
\(945\) 40.5845 1.32022
\(946\) 0 0
\(947\) 29.2710 0.951181 0.475591 0.879667i \(-0.342234\pi\)
0.475591 + 0.879667i \(0.342234\pi\)
\(948\) 0 0
\(949\) −45.5964 −1.48012
\(950\) 0 0
\(951\) −4.16799 −0.135156
\(952\) 0 0
\(953\) −42.8612 −1.38841 −0.694205 0.719778i \(-0.744244\pi\)
−0.694205 + 0.719778i \(0.744244\pi\)
\(954\) 0 0
\(955\) 13.5668 0.439011
\(956\) 0 0
\(957\) 6.91540 0.223543
\(958\) 0 0
\(959\) 2.96669 0.0957995
\(960\) 0 0
\(961\) −8.86817 −0.286070
\(962\) 0 0
\(963\) −5.28871 −0.170426
\(964\) 0 0
\(965\) −34.3763 −1.10661
\(966\) 0 0
\(967\) −26.3728 −0.848093 −0.424046 0.905640i \(-0.639391\pi\)
−0.424046 + 0.905640i \(0.639391\pi\)
\(968\) 0 0
\(969\) −15.3176 −0.492071
\(970\) 0 0
\(971\) 48.7680 1.56504 0.782520 0.622625i \(-0.213934\pi\)
0.782520 + 0.622625i \(0.213934\pi\)
\(972\) 0 0
\(973\) 74.9226 2.40191
\(974\) 0 0
\(975\) −15.2013 −0.486831
\(976\) 0 0
\(977\) 44.0831 1.41034 0.705172 0.709037i \(-0.250870\pi\)
0.705172 + 0.709037i \(0.250870\pi\)
\(978\) 0 0
\(979\) 42.4190 1.35572
\(980\) 0 0
\(981\) −2.37466 −0.0758170
\(982\) 0 0
\(983\) −44.6649 −1.42459 −0.712294 0.701881i \(-0.752344\pi\)
−0.712294 + 0.701881i \(0.752344\pi\)
\(984\) 0 0
\(985\) 1.09519 0.0348958
\(986\) 0 0
\(987\) −58.7989 −1.87159
\(988\) 0 0
\(989\) 30.1419 0.958457
\(990\) 0 0
\(991\) −25.0263 −0.794985 −0.397493 0.917605i \(-0.630120\pi\)
−0.397493 + 0.917605i \(0.630120\pi\)
\(992\) 0 0
\(993\) −15.3905 −0.488404
\(994\) 0 0
\(995\) −20.2540 −0.642096
\(996\) 0 0
\(997\) 22.8643 0.724118 0.362059 0.932155i \(-0.382074\pi\)
0.362059 + 0.932155i \(0.382074\pi\)
\(998\) 0 0
\(999\) 64.5704 2.04292
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4016.2.a.m.1.16 23
4.3 odd 2 2008.2.a.d.1.8 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.8 23 4.3 odd 2
4016.2.a.m.1.16 23 1.1 even 1 trivial